Product Choice and Product Switching
Andrew B. Bernard, Stephen Redding and Peter K. Schott
evidence we present on the importance of product switching by U.S. manufacturers. Two-thirds of continuing firms change their product mix every five years, and product switches involve more than 40% of firm output and almost half of existing products. The theoretical model incorporates heterogeneous firms, heterogeneous products, and ongoing entry and exit. In equilibrium, firm productivity is correlated with product fixed costs, with the most productive firms choosing to make the products with the highest fixed costs. Changes in market structure result in systematic patterns of firm entry/exit and product switching.
Keywords: heterogeneous firms, product differentiation, sunk costs, entry and exit JEL Classification: L11, D21, L60
The Centre for Economic Performance is financed by the Economic and Social Research Council. This paper was produced as part of the Centre’s Globalisation Programme.
Acknowledgements
We thank Jonathan Haskel, Marc Melitz, Danny Quah, and seminar participants at CEPR, Dartmouth, Penn State, and LSE for helpful comments. Andrew B. Bernard and Peter K. Schott gratefully acknowledge research support by the National Science Foundation. Stephen Redding gratefully acknowledges financial support from a Philip Leverhulme Prize . The research in this paper was conducted at the Center for Economic Studies. Research results and conclusions expressed are those of the authors and do not necessarily indicate concurrence by the Bureau of the Census or by the National Bureau of Economic Research. The paper has not undergone the review the Census Bureau gives its official publications. It has been screened to insure that no confidential data are revealed.
Andrew B. Bernard is Tuck School of Business at Dartmouth and National Bureau of Economic Research. Stephen Redding is London School of Economics and Centre for Economic Policy Research. Peter K. Schott is Yale School of Management and National Bureau of Economic Research
Authors addresses:
Andrew B. Bernard Stephen Redding Peter K. Schott
Tuck School of Business at Dartmouth CEP and Department of Economics Yale School of Management 100 Tuck Hall London School of Economics 135 Prospect Street
Hanover Houghton Street New Haven
NH 03755 London WC2A 2AE CT 06520
email: [email protected] email: [email protected] email: [email protected]
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A. B. Bernard, S. Redding and P. K. Schott, submitted 2003 ISBN 0 7530 1672 9
1. Introduction
The choice of which product to make is a fundamental decision by a firm. Despite this, most theoretical models of firm behavior take product ch o i c e by t h e firm as give n, or treat entry i nto a pro duct marke t t o b e t he same as the decision to create a firm. Similarly, there is no systematic evidence on the importance of product switching for the firm or on the scope of product switching across industries.
This paper models product choice by heterogeneous firms. The theory is motivated by a new set of stylized facts where we document the im-portance of product switching across U.S. manufacturing firms. Product switching, i.e. the addition or deletion of a product from the firm’s output mix, is a widespread activity. More than two thirds of continuing U.S. manufacturing firms alter their product mix over a five year period. For firms, adding or dropping a product is an important event, affecting more than one third of current products and current output. Over half of all product additions also expand the mix of industries produced by the firm. Our theoretical model incorporates heterogeneous firms, heterogeneous products, and ongoing entry and exit in general equilibrium. Firms choose which product to produce within an industry as well as whether to enter or exit the market. Products differ both in terms of how they are de-manded and in their production techniques. Product choice is determined by an interaction between firm characteristics, product characteristics, and market conditions. Changes in market characteristics, production technolo-gies, or consumer tastes are associated with changes in entry and exit rates, industry and product productivity, and product switching by firms.
We model firms as being heterogeneous in terms of their productivity and products as differing in terms of their fixed costs of production. In equilibrium, firm productivity is correlated with product choice, with the most productive firms endogenously choosing to make the products with the highest fixed costs. Both product choice and entry/exit decisions are endogenously determined by market structure.
Our framework builds on Melitz’s (2002) dynamic industry model with heterogenous firms. We extend that model by introducing multiple prod-ucts defined by their fixed and variable costs of production and their sub-stitutability in consumption. Our framework incorporates endogenous firm entry and exit decisions and variation in firm productivity (as in Hopenhayn
1992 and Jovanovic 1982); horizontal product differentiation and monop-olistic competition (as in Krugman 1980); and endogenous product choice based on production technique or quality (as in Shaked and Sutton 1987 and Sutton 1998).
This paper concentrates on interactions of firm characteristics, prod-uct characteristics and market conditions as sources of prodprod-uct switching. However, product choice is not only interesting in its own right, but po-tentially has implications for the firm’s decisions about investment, factor and material purchases, and pricing that lie beyond the scope of this pa-per. Similarly for the industry, we consider the relation of product choice by firms to market structure and productivity growth and leave unmod-elled consequences for international trade and investment flows, returns to factors, and the nature of competition.
The remainder of the paper is structured as follows. Section 2 provides evidence on product switching by U.S. manufacturing firms from 1972-1997. Section 3 considers existing empirical and theoretical work on industries with heterogeneous firms and presents a non-technical overview of the the-oretical model. In section 4, we develop the model while section 5 solves for general equilibrium. A technical appendix at the end of the paper contains important derivations and proofs of the propositions. Section 6 examines the effects of changes in the model’s parameters on firm-level choices and industry equilibrium. Section 7 concludes.
2. Product Switching By U.S. Manufacturing Firms, 1972-1997 In this section we provide the first complete view of product switching by U.S. manufacturing firms.1 We look at the importance of adding and
1Dunne, Roberts, and Samuelson (1988) use U.S. Census data from 1963-1982 to
examine forms of entry into industries by firms. They find that just under half of all new entrants in an industry come from continuing firms, i.e. firms adding an industry. Their paper examines rates of (more aggregate) industry entry and exit and subsequent performance by different types of entrants rather than the extent and importance of product additions and deletions by firms. Streitweiser (1992) uses U.S. Census data to describe the number of products (SIC5) produced by U.S. manufacturing plants during the period 1972-1982. Her focus is on the similarity of products produced at multiple product plants, not on product switching. Gollop and Monahan (1991) also use U.S. Census data to construct an index of firm diversification based on 5-digit products. They report increasing firm diversification and declining plant diversification within most
2-dropping products for the manufacturing sector a whole as well as for in-dividual firms. We start by documenting the extent of this activity across firms in the U.S. manufacturing sector. Next, we consider the importance of product-switching at the firms themselves. Finally, we ask whether product switches are associated with significant changes in the firm’s out-put mix.
Our data comes from the Censuses of Manufactures (CM) of the Longi-tudinal Research Database (LRD) of the U.S. Bureau of the Census starting in 1972 and conducted every fifth year through 1997. The sampling unit for the Census is a manufacturing establishment, or plant, and the sam-pling frame in each Census year includes information on inputs, output, and products on all establishments. We aggregate the plant-level data up to the enterprise, or firm, for all the results reported in this paper. Ex-amining the product mix of the firm, rather than the plant, has several advantages: it is both the level at which decisions are made about prod-ucts and it avoids a potential problem of a firm shuffling its existing mix of products across plants. From the Census, we construct firm characteristics including the total value of shipments, the number of products produced, total value of each product produced, and information on the births and deaths of firms.
In constructing our sample, we make several modifications to the basic data. We use information on all manufacturing establishments in the six Censuses. While the LRD does contain basic information on small plants (so-called Administrative records), we do not include them in this study due to the lack of information on products. We aggregate the establishment level data in the Censuses up to the level of the firm. All our results are based on firm-level statistics. On average we are left with 141,561 continuing firms in each year.2
We refer to a product as a unique five-digit category in the 1987 Stan-dard Industrial Classification (SIC5).3 In Census years, plant output is
digit industries from 1963-1982 but do not address issues related to product switching. Bernard et al. (2002) provide evidence of U.S. manufacturing firms switching between industries in response to competition from low-wage countries.
2On average, over one third of manufacturing firms do not survive from one Census
to the next. In these tables, we focus on product-switching at surviving, or continuing, firms. The subsequent theoretical model allows for firm failure as an outcome.
3
recorded either at the five-digit or seven-digit SIC level of detail. Roughly 7,000 of the 15,000 seven-digit categories are recorded directly in the LRD, the rest are recorded at a five-digit level of aggregation. We aggregate seven-digit categories up to their five-digit ‘product-class’ to obtain a com-plete set of SIC5 products for all manufacturing firms. Our terminol-ogy differs slightly from that of the Census Bureau. We [Census] define four-digit categories as industries [industries] and five-digit categories as products [product classes].
Table 1: Average Share of US Manufacturing Firm Activity During 5-Year Census Period Intervals, 1972 to 1997
Type of Firm All Firms Single Product
Firms
Multiple Product Firms
Firm takes no action 31 47 11
Firm drops products only 13 na 31
Firm adds products only 12 15 8
Firm both adds and drops products 43 38 50
Type of Firm All Firms Single Product Firms Product FirmsMultiple
Firm takes no action 7 51 3
Firm drops products only 8 na 8
Firm adds products only 5 21 3
Firm both adds and drops products 81 29 86
Note: Table displays average share of continuing US manufacturing firms engaging in each activity across five year intervals from 1972 to 1997. Add and Drop refer to firms adding or dropping at least one five-digit SIC manufacturing 'product' during the five years between Censuses of Manufactures. The upper panel reports the distribution of firms while the lower panel reports the distribution of output. Results are reported for the full sample of firms as well as for single- and multiple-product firms separately. On average, there are roughly 140,000 firm observations in each Census year. "na" indicates not applicable: single product surviving firms cannot drop their only product and continue in business.
Percent of Firms
Percent of Output
Product switching is a pervasive activity among U.S. manufacturing
switching using Standard Industrial Classification codes. This has a number of im-portant advantages: firms report several SIC5 codes and we can explicitly observe the addition and deletion of products; also, SIC5 codes are typically chosen on the basis of distinct product characteristics and are more aggregated than varieties of the same product. See US Census (1996), http://www.census.gov/prod/2/manmin/mc92-r-1.pdf, for a complete list of SIC5 categories.
firms. During a typical five year period, two thirds of continuing firms add or drop a (SIC5) product from their output mix.4 This activity is
summarized in the first panel of Table 1, which reports a breakdown of product switching for continuing U.S. manufacturing firms. The numbers in the table refer to the average fraction of firms undertaking the activity described in the first column across five year Census intervals from 1972 to 1997. The numbers reported in all the tables are averages from 1972-1997, there is little variation across Censuses. We divide all continuing firms into one of four types: (a) those that leave their product mix unchanged, i.e. neither add nor drop a SIC5 product; (b) those that drop at least one product and do not add any; (c) those that add at least one product but do not remove an existing product from their output mix; and (d) firms that both add and drop at least one product. The first four rows of the table summarize these four mutually exclusive activities.
Every five years, 43% of continuing firms shuffle their product mix by both dropping and adding products (column 1 of Table 1). Dropping alone and adding alone are far less frequent events, occurring at 13% and 12% of firms respectively.
The second and third columns of Table 1 compare the activity of sin-gle versus multiple-product firms. Sinsin-gle-product firms are likely to leave their product mix unchanged (47%) or change it completely by dropping their existing product and adding one or more new ones (38%). Half of all multiple-product firms both add and drop products, while 31% of such firms narrow their product mix by dropping one or more products. On average 89% of continuing multiple-product firms alter their product mix during a five year period. These numbers clearly demonstrate that product-switching is widespread among U.S. manufacturers and that si-multaneous additions to and deletions from the existing product mix are
4A product addition in the data can represent one of three general activities. The first,
related to product innovation, is the creation of an entirely new good at an existing plant which is classified in an existing SIC5 category. The second is the start of production of an existing good, new to the firm but not to the market, also in an existing SIC5 category. Innovation of a new good can occur but not be captured as a product addition if it occurs in a SIC5 category where the firm is currently active. Finally, a product addition may represent the firm’s acquisition of a plant that is producing one or more products new to the firm. The category of product additions through acquisitions accounts for fewer than 10% of all switches.
frequent events.
While product-switching occurs at a large proportion of firms it is even more important in terms of output. The second panel of Table 1 shows the fraction of total output produced by the four firm types. More than 80% of total output is generated at firms that both add and drop products. Multi-product firms that are large in size are the most active in changing their product mix in both directions.
Table 2: Average Number of Products Produced, Added and Dropped Across 5-Year Intervals, 1972 to 1997
Type of Firm Number of Products Number of Adds Number of Drops All Firms 2.3 1.1 1.1
Single Product Firms 1.0 0.8 0.4
Multiple Product Firms 4.0 1.4 2.0
Note: Table displays mean number of products produced by continuing US manufacturing firms across Census of Manufacturing years 1972-1997. Adds and Drops refer to the number of five-digit SIC
manufacturing products added to or dropped from a firm's product mix during the five years between censuses. Results are reported for the full sample of firms as well as for single- and multiple-product firms separately.
Product turnover among manufacturing firms is substantial and occurs at both single-product and multiple-product firms. Table 2 reveals that the average continuing U.S. manufacturing firm produces 2.3 products at the beginning of a five year period, and subsequently adds and drops almost half of its product mix (1.1 products). On average, single-product firms add products while on average multiple-product firms drop half their existing products (2 dropped out of 4 produced) and add 1.4 new products.5
The products that are added and dropped by switching multi-product firms account for a large fraction of firm output.6 On average, products that were added in the last five years account for 50% of current firm output by value. Products that will be dropped over the next five years represent 46% of current shipments, Table 3. For the median firm, recently added products are 42% of shipments while products to be dropped are more than
5
Over time from 1972-1997, there is no systematic change in the number of products produced by continuing plants, nor in the fraction of products added or dropped.
6
Table 3: Average Share of Firm Output in Products Added and Dropped in Multi-Product Firms Across 5-Year Intervals, 1972 to 1997
Activity Number of Firms Mean Share of Firm Output Median* Share of Firm Output Added Products 150,854 50 42 Dropped Products 142,178 46 34
Notes: Table displays mean and median value of continuing US
manufacturing firm's added and dropped products as a percent of total firm output. Products are five-digit SIC aggregates. Figures are based on all adding and dropping behavior across five-year Census of Manufacturing years 1972 to 1997. Because of Census disclosure rules, which prohibit the reporting of an actual median, the reported median is the average of the 100 observations around the true median.
a third of output by value.
The previous facts demonstrate that product-switching is a widespread phenomenon in the U.S. manufacturing sector, occurring at more than two thirds of continuing firms that produce more than 90% of total output. In addition, the results reveal that a sizable fraction of products are turned over every five years and that these products represent a large share of existing firm output. However, there remains the possibility that such activity involves switches among products that are quite similar. For example, switches between Canned Fruits and Canned Vegetables, both of which are five-digit products in SIC 2033, are unlikely to indicate important economic changes at a firm (see Appendix A for further discussion of the data).
To identify whether firms are merely adding products that are similar in nature, we examine the fraction of added-products that extend the range of production of the firm across industries.7 We focus our attention on product additions in multi-product firms. The first column of Table 4 shows that 54% of all added products are outside existing four-digit (SIC4) industries produced by the firm. Furthermore, more than a third of all new products are in a new three digit (SIC3) industry while 17% of added products involve a new two digit (SIC2) industry. The second column of the same table reports similar percentages for firms. Of firms that add
7Almost all empirical work on U.S. manufacturing identifies an “industry” by its
Table 4: Share of Product Additions Outside the Existing Industry Mix of the Firm, 1972 to 1997
(I) (II)
Share of added products in: Share of firms adding a product(s) that:
existing SIC4 industries 46 do not add a SIC4 industry 42
new SIC4 industries 54 add a new SIC4 industry 58
new SIC3 industries 35 add a new SIC3 industry 41
new SIC2 industries 17 add a new SIC2 industry 22
Notes: Table reports the extent to which firms add products outside their existing mix of industries. The first column of the table reports the share of added five-digit SIC products that are outside of the firm's existing stable of four-, three- and two-digit SIC manufacturing industries. The second column of the table reports the share of firms adding at least one five-digit SIC manufacturing product outside its existing mix of four-, three- and two-five-digit manufacturing industries.
Product Additions and Changes in Firm Industry Mix
one or more products, 58% also add a new four-digit industry, while 41% (22%) add a new three (two) digit industry to the mix produced by the firm. These results confirm that product-switching involves substantial changes in industry composition and tends to move a firm well beyond its existing mix of industries.
This section has documented the extent and quantitative importance of product switching. The remainder of the paper develops a theoretical model that explicitly incorporates product switching, while also remain-ing consistent with existremain-ing empirical work emphasizremain-ing heterogeneous firm characteristics and continuing entry and exit. We develop a general equi-librium dynamic industry model that, for the first time, simultaneously explains these three phenomena.
3. Many Firms and Multiple Products
The facts in the previous section suggest that product choice at surviv-ing firms should be added to the growsurviv-ing list of stylized facts about industry performance. From Dunne, Roberts, and Samuelson (1988, 1989) we know that there is substantial covariation in industry entry and exit rates, sug-gesting an important role for sunk costs of entry. From Dunne, Roberts and Samuelson (1988, 1989), Davis and Haltiwanger (1991), Bernard and Jensen (1995) and the survey by Bartelsman and Doms (2000), we know that there is an enormous amount of firm heterogeneity within narrowly
classified industries and that firm characteristics, such as productivity, are correlated with exit/survival and entry into exporting.
A series of theoretical papers have built upon these stylized facts in mod-elling industry evolution and performance. Jovanovic (1982) and Hopen-hayn (1992) incorporate firm heterogeneity in equilibrium industry models with covariation in entry and exit rates due to underlying variation in sunk costs. Both models, and Olley and Pakes (1996), have the feature that firms will exit when their productivity falls below an endogenously determined cutoff. Bernard et al (2003) and Yeaple (2002) model the interaction of firm heterogeneity and entry into exporting. Melitz (2002) adds imperfect competition to a version of Hopenhayn (1992), allowing for industry en-try and exit rates to covary and for firm characteristics to matter for the decision to export.
Our theoretical model extends a version of the heterogeneous firm frame-work of Melitz (2002) and Hopenhayn (1992) to incorporate product choice by surviving firms. We explicitly build on existing links between theory and stylized facts about firm heterogeneity and industry exit and entry rates. The key contribution of our model is to allow firms to choose endogenously between distinct products. At the heart of the model lies an interaction between firm heterogeneity and product heterogeneity that shapes product choice and product switching.
The model portrays an industry with a number of distinct products each containing a continuum of horizontally differentiated varieties.8 The prod-ucts are imperfect substitutes and have different production technologies, specifically the fixed and variable costs. Each firm chooses to produce a variety of one of the two products. In addition to product choice, the model incorporates firm births and deaths, imperfect competition, and firm het-erogeneity.
The model allows us to examine how changes in market conditions affect product choice through the interaction of firm characteristics and product attributes. Endogenous product switching by firms affects industry pro-ductivity and firm profitability. For example, changes in the sunk cost of starting a firm (entry cost) affect the distribution of firm productivity and,
8
The model is for any ‘market’ with a common sunk cost of entry. Given the evidence on the variation of sunk costs across industries we refer to the market as an industry with heterogeneous products.
at the same, induce some firms to switch products.9
The demand structure of the model is quite simple. Representative consumers have a taste for both products and have a love of variety within each product. Products are less substitutable in consumption than are varieties. Each variety is produced with a single input, labor, and is subject to a fixed cost (repeated each period) and a constant variable cost. A key feature of the model is that the period-by-period fixed costs vary across products (but not across varieties within products). Variable costs are also identical for all varieties but may or may not differ across products.10 This provides a particularly tractable way of formalizing the more general idea that there are differences in production technologies across products.
One interpretation of the difference in fixed costs across products is that it captures a difference in physical capital intensity. Although labor is the sole factor of production in the model, the labor used in paying the product fixed cost can be thought of as labor set aside to build and, in subsequent periods, maintain a machine. Another interpretation is that the difference in fixed costs captures ongoing expenditures that allow one product to be a higher quality than another. In this case, since the variable cost of production can be thought of as an inverse measure of product quality, the high fixed cost product must be characterized by a lower variable cost of production (higher product quality). More generally, as noted above, we make no assumptions about the relative value of the variable cost of production for the two products.
For tractability, the theoretical work focuses on the case of a single industry whose two products enter directly into final consumption. It is relatively straightforward to introduce many industries. The framework
9
While we focus on the interaction of firm and product characteristics in a dynamic industry setting, we recognize that there are a variety of additional reasons that firms may add or drop products. Classic treatments of product choice include Hotelling (1929), Chamberlain (1951) and Lancaster (1966). More formal treatments of horizontal differentiation and vertical differentiation include Dixit and Stiglitz (1977), Shaked and Sutton (1982, 1987) and Spence (1976), synthesized in Tirole (1988). Issues related to product choice can also be found in work on corporate diversification, mergers and divestitures, and evolving firm capabilities and factor accumulation. See, among others, Amihud and Baruch (1981), Bolton and Farrell (1990), Chandler (1990), Helfat and Raubitschek (2000), Milgrom and Roberts (1990), and Montgomery (1994).
1 0Yeaple (2002) allows technologies for producing varieties to vary in terms of both
can also be extended to allow for any number of products within the in-dustry. These extensions merely complicate the analysis without changing the model’s key insights.11
There is a large pool of ex-ante identical potential entrants into the industry. To enter the industry (and hence either product market within the industry), firms must pay a sunk cost of entry. Firms then observe their productivity, drawn from a common distribution. At this point each firm decides whether to produce or exit the market. Firms with low pro-ductivity draws immediately exit. If the firm decides to produce it chooses to manufacture either product 1 or product 2 depending on the relative profitability of each product for that firm given its observed productivity. All firms, regardless of the product they produce, face a common exogenous probability of death in every period which, for simplicity, is assumed not to depend on their productivity.12
The solution of the model depends on three key equilibrium conditions: the zero profit cutoff condition which identifies the lowest productivity firm that chooses to produce; the product indifference cutoff condition which identifies the productivity level at which a firm is exactly indifferent between producing either of the two products13; and the free entry con-dition which forces expected profits before entry to equal the sunk cost of entry and is driven by the existence of a competitive fringe.
In the next three sections, we introduce the theoretical model in further detail, characterize general equilibrium, and explore how firms’ product choice and the distribution of productivity across firms responds to changes in the underlying parameters of the model.
1 1For simplicity, we assume that all products already exist, although new varieties are
created when the number of producing firms increases.
1 2This assumption supports a particularly tractable model of product choice,
en-try/exit, and firm heterogeneity. See Hopenhayn (1992) for an analysis of industry dynamics where productivity affects the probability of firm death, but that abstracts from product choice.
1 3A firm with this productivity would be either the highest productivity firm in the
4. Theoretical Model: Endogenous Product Choice and Industry Dynamics
The preferences of a representative consumer are a nested CES function of consumption of varieties of two products:
U = [aC1ν+ (1 − a)C2ν]1/ν. (1) where Ci is a consumption index of a continuum of horizontally
differenti-ated varieties ω in each product market i:
Ci = ·Z ω∈Ωi qi(ω)ρdω ¸1/ρ (2)
and {Ωi} is the mass (number) of available varieties in market i.
On the demand side, products are distinguished from varieties in terms of consumers’ elasticity of substitution. The two products are imperfect substitutes for one another, with elasticity ψ = 1−ν1 > 1. Similarly, vari-eties of each product are imperfect substitutes, with elasticity σ = 1−ρ1 > 1. However, the elasticity of substitution between varieties of the same product is higher than that between different products: σ > ψ > 1. The parameter 0 < a < 1 is a demand-shifter, which captures the relative weight of the two products in the consumer’s utility function.
The dual price index for each product is:
Pi= ·Z ω∈Ωi pi(ω)1−σdω ¸1/1−σ . (3)
and expenditures over varieties of each product are given by:
ri(ω) = Ri µ pi(ω) Pi ¶1−σ = αi(P) R µ pi(ω) Pi ¶1−σ (4) where R = R1+ R2 =Rω∈Ω 1r1(ω)dω + R ω∈Ω2r2(ω)dω is aggregate
expen-diture; P = P2/P1 is the relative price of product 2; and αi(P) = Ri/R
is the share of expenditure devoted to product i. With CES utility, these expenditure shares are:
α1(P) = " 1 + µ 1 − a a ¶ψ P1−ψ #−1 , α2(P) = 1 − α1(P) . (5)
4.1. Production
Each of a continuum of firms produces a unique variety ω in one of the two product markets. Production requires one factor, labor, which is inelastically supplied at its aggregate level L , which also indexes the size of the economy.
As well as entering demand as imperfect substitutes, the two products have different production technologies. The fixed costs of production for the two products are given by {f1, f2} where f2 > f1. For firms of the same
productivity, ϕ, bi is a parameter that indexes variable costs of production
for product i where bi > 0. The key distinction between products is that
the second good is subject to a higher fixed cost of production. We allow the variable cost for product 2 to be either higher or lower than for product 1. Without loss of generality, we set b1 = 1 and b2= b, so that b captures
the relative value of the two products’ variable costs.
Given firm productivity ϕ, labor used is a linear function of output q: li = fi + biϕqi. Profit maximization under imperfect competition yields
the standard result that equilibrium prices are a constant mark-up over marginal cost, pi(ϕ) = µ σ σ − 1 ¶ wbi ϕ . (6)
We choose the wage as numeraire so that w = 1. Equilibrium profits for the two products are thus,
πi(ϕ) = µ αi(P)R σ ¶ µ Piρ ϕ bi ¶σ−1 − fi (7) πi(ϕ) = ri(ϕ) σ − fi.
From firms’ revenue functions, the ratio of revenues for any two firms producing the same product depends only on their relative productivities,
ri(ϕ00) ri(ϕ0) = µ ϕ00 ϕ0 ¶σ−1 . (8)
straightforward; the ratio of revenues is a function of their relative produc-tivities and relative prices of the two products,
r2(ϕ00) r1(ϕ0) = µ 1 − α1(P) α1(P) ¶ ·µ ϕ00 ϕ0 ¶ P1b ¸σ−1 . (9)
where we have used b1 = 1 and b2 = b. These relationships are useful
because they allow us to express the revenue earned by every firm in terms of the revenue earned by another firm with a particular level of productivity.
4.2. Market Entry and Exit
To produce a variety (of either product), firms must pay a common fixed entry cost, fe > 0, which is thereafter sunk. After paying the sunk
cost, firms draw their productivity, ϕ, from a common distribution, g (ϕ). Firm productivity remains fixed after entry, i.e. we assume there is no feedback from product choice to productivity. Upon entering, there is an exogenous probability of death each period, δ, which we interpret as due to force majeure events beyond managers’ control.
The value of a firm with productivity ϕ is therefore,
v (ϕ) = max ( 0, ∞ X t=0 (1 − δ)tπ1(ϕ) , ∞ X t=0 (1 − δ)tπ2(ϕ) ) = max ½ 0,1 δπ1(ϕ) , 1 δπ2(ϕ) ¾ (10)
4.3. Endogenous Product Selection
Firms decide which good to produce based upon their productivity, taking as given the aggregate price indices {P1, P2}. Firms with zero
productivity have negative post-entry profits in both industries, with the loss greatest for the high fixed cost product 2:
π1(0) = −f1 > π2(0) = −f2. (11)
A firm with productivity ϕ will produce good i if it brings positive profit, i.e. covers the fixed production cost fi, and is more profitable than
producing good j. For both goods to be produced in equilibrium, it is sufficient for π1(ϕ) > π2(ϕ) > 0 for some values of ϕ and π2(ϕ) > π1(ϕ) > 0
for other values of ϕ. Via (11), these conditions require product 1 profits to increase with productivity (dπ1(ϕ)
dϕ > 0) and for product 2 profits to increase
faster than product 1 profits (dπ2(ϕ)
dϕ > dπ1(ϕ)
dϕ ) for at least a range of values
for ϕ. The second of these conditions is met if and only if dπ2/dϕ dπ1/dϕ = µ 1 − a a ¶ψµ 1 b ¶σ−1 Pσ−ψ > 1 (12)
where we have used the expression for the CES expenditure share, α1(P),
from equation 5. This inequality does not depend on ϕ, but rather on relative prices and parameters of demand and technology.
Two profit curves meeting the conditions for both products to be pro-duced are displayed in Figure 1. The curve for π2 intersects π1 from below
in the region where positive profits are made. Firms with productivity below ϕ∗ exit immediately upon entering the market because their produc-tivity is too low to earn positive profit in either product market. Firms with productivity between ϕ∗ and ϕ∗∗ enter the low fixed cost product market while firms with productivity greater than ϕ∗∗enter the high fixed cost product market.14
Figure 1 provides intuition for two key equilibrium conditions that we use to solve the model. First, the zero profit cutoff condition deter-mines the lowest level of productivity where product 1 is produced and is given by,
(ZP) π1(ϕ∗) = 0. (13)
Second the product indifference cutoff condition determines the lowest level of productivity where product 2 is produced and is given by,
(PI) π2(ϕ∗∗) = π1(ϕ∗∗) . (14)
4.4. Free Entry
A third equilibrium condition, the free entry condition, equates ex-pected firm value and entry costs and is driven by the assumed existence
1 4
Of course, in order for both goods to be produced, we also require that the fixed costs of production are not large enough to exhaust the economy’s entire supply of labor. This condition must be satisfied as f1→ 0 and f2→ f1.
ϕ
π
-f
1-f
2ϕ
∗ϕ
∗∗π
1π
2Figure 1: Profit Versus Productivity for the Two Products
of a competitive fringe of potential entrants. The expected value of a firm before entry, ve, consists of two components. The first is the ex ante
prob-ability of producing product 1 times the expected profitprob-ability of producing this good until death. The second is the ex ante probability of produc-ing product 2 times the expected profitability of producproduc-ing this good until death. The cost of entry is equal to the sunk cost, fe.
Using the definitions of the productivity cutoffs (ϕ∗, ϕ∗∗) above, the free entry condition is given by:
(FE) ve=
G (ϕ∗∗) − G (ϕ∗)
δ π1+
1 − G (ϕ∗∗)
δ π2= fe, (15) where π1 and π2 are the expected or average profits from producing
prod-ucts 1 and 2, respectively. In Appendix B, we show that average profit in each market is equal to the profit earned by a firm with weighted av-erage productivity: π1 = π1(ϕ) and πe 2 = π2(eϕ), wheree ϕ and ee ϕ are thee
weighted average productivities of firms manufacturing products 1 and 2 In addition, weighted average productivity is a function of the two cutoff productivity levels, ϕ∗ and ϕ∗∗.
4.5. Goods and Labor Markets
The steady-state equilibrium is characterized by a constant mass of firms entering each period Me, a constant mass M1 producing product 1,
and a constant mass M2 producing product 2.
As shown in Appendix B, the product prices (PP) may be written as:
(PP) P1 = M11/1−σp1(ϕ),e P2 = M21/1−σp2(eϕ)e (16)
where p1(ϕ) and pe 2(eϕ) are the prices charged by a firm with weightede
average productivity in each market.
A condition for steady-state equilibrium is that the mass of firms who enter and decide to produce a product equals the mass of current produc-ers who die. Each period, a fraction δ of firms die. Thus, from the definitions of the productivity cutoffs above, these steady-state stability conditions(SC) are:
(SC) [1 − G(ϕ∗∗)]Me = δM2 (17)
[G(ϕ∗∗) − G(ϕ∗)]Me = δM1. (18)
The labor market clearing condition (LM), expressed in value terms, requires that total labor payments equal the sum of payments to labor used in production and payments to labor used in the sunk costs of entry:
(LM) L = Lp+ Le (19)
where, since labor has been chosen for the numeraire, w = 1.
5. Equilibrium
An equilibrium is referenced by the sextuple, {ϕ∗, ϕ∗∗, P1, P2, R1, R2}:
all other endogenous variables may be written as functions of these vari-ables. Equilibrium is fully characterized by the zero profit cutoff condi-tion, the product indifference cutoff condicondi-tion, the free entry condicondi-tion, the steady-state stability condition (constant aggregate variables), the labor market clearing condition, and the product prices condition.
5.1. Product Supply and Relative Prices
Combining the zero-profit ZP and product indifference PI conditions, we arrive at a product supply relationship between the productivity cut-offs (ϕ∗∗/ϕ∗) and relative prices (P) that must hold when both goods are produced: ϕ∗∗ ϕ∗ ≡ Λ = ³ f2 f1 − 1 ´ h¡1−a a ¢ψ¡1 b ¢σ−1 Pσ−ψ− 1i 1/(σ−1) (20)
where, in order for both goods to be produced, we require: µ f2 f1 − 1 ¶ > "µ 1 − a a ¶ψµ 1 b ¶σ−1 Pσ−ψ− 1 # > 0. (21)
As discussed below, relative prices adjust to ensure that this condition is satisfied in equilibrium. The middle inequality is just the requirement that profits for product 2 increase more rapidly in productivity than profits for product 1 (equation (12)), and ensures the right-hand side of (20) is positive. The first inequality incorporates information on fixed production costs, and ensures that profits for product 2, π2, intersect profits for product
1, π1, from below in the region where positive profits are made (Figure 1),
so that the right-hand side of (20) is strictly greater than 1.
Equation (20) provides a downward sloping relationship between the productivity cutoffs, ϕ∗∗/ϕ∗, and relative prices, P. As the relative price of product 2 rises, this raises the profitability of product 2 relative to product 1, inducing firms to switch from product 1 to product 2, and leading to a fall in ϕ∗∗ relative to ϕ∗.
5.2. Product Demand and Relative Prices
Combining the expression for product prices PP with the steady-state stability conditions SC, we arrive at a product demand relationship be-tween the productivity cutoffs (ϕ∗∗/ϕ∗) and relative prices (P):
"Rϕ∗∗ ϕ∗ ϕσ−1g (ϕ) dϕ R∞ ϕ∗∗ϕσ−1g (ϕ) dϕ # = µ P b ¶σ−1 . (22)
Since the left-hand side is monotonically increasing in ϕ∗∗/ϕ∗, equation (22) implies an upward sloping relationship between ϕ∗∗/ϕ∗and P. As the
relative price of product 2 rises, this reduces demand for product 2 relative to product 1, inducing consumers to switch from product 2 to product 1, and leading to a rise in ϕ∗∗ relative to ϕ∗.
Combining the downward sloping (20) and the upward sloping (22) defines a unique equilibrium pair (P, ϕ∗∗/ϕ∗), which satisfies the inequality in equation (21) so that both products are produced.15
Starting from equilibrium, if there is a rise in the relative price for product 2, P, there are two effects. On the supply side, a rise in P reduces ϕ∗∗ relative to ϕ∗, while on the demand side, it increases ϕ∗∗ relative to ϕ∗. Since relative supply of product 2 has increased and relative demand has fallen, this cannot be an equilibrium and P must fall in order to bring relative supply and demand back into balance.
5.3. Free Entry
The free entry condition FE equating expected firm value and entry costs depends on average profitability in each market, which itself depends on the two productivity cutoffs (ϕ∗, ϕ∗∗) and relative prices P. Using the relationship between the productivity cutoffs in (20), Appendix B shows that FE may be written as:
ve= f1 δ Z Λϕ∗ ϕ∗ "µ ϕ ϕ∗ ¶σ−1 − 1 # g(ϕ)dϕ (23) +f1 δ Z ∞ Λϕ∗ "µ 1 − a a ¶ψµ1 b ¶σ−1 Pσ−ψ µ ϕ ϕ∗ ¶σ−1 − f2 f1 # g(ϕ)dϕ = fe.
This provides another equation, which together with (22) and (20), de-termines unique equilibrium values of three unknowns (ϕ∗, ϕ∗∗, P). The
equilibrium values (ϕ∗, ϕ∗∗, P) are sufficient to determine weighted average productivity and average profitability in each market, as shown in Appen-dix B.
1 5
5.4. Goods and Labor Markets
Total payments to labor used in production equal the difference between total revenue, R, and total firm profits, Π: Lp = R − Π. Combining SC
and LM, total payments to labor used in entry are exactly equal to total firm profits, Le= Mefe= Π, reflecting the existence of a competitive fringe
of firms entering until the expected value of entry exactly equals the sunk cost. Together, these results imply that total revenue, R, equals total payments to labor, L, and the labor market clears.
Revenue in each product market may be determined from the CES ex-penditure share (equation (5)) at equilibrium relative prices: R1 = α1(P)R
and R2= (1 − α(P))R.
The absolute levels of the price indices, P1 and P2, in equation (16)
depend on the mass of firms producing each product, M1 = (R1/r1) and
M2 = (R2/r2), where aggregate revenue in each product market was
de-termined above and average revenue can be written as a function of the productivity cutoffs and relative prices alone (ϕ∗, ϕ∗∗, P).
Proposition 1 There exist unique equilibrium values of {ϕ∗ , ϕ∗∗ ,P1, P2,
R1, R2} that solve the zero profit, product indifference, free entry,
steady-state stability, labor market clearing, and product prices conditions
Proof. See Appendix B
Proposition 2 (Aggregate Variables)
The equilibrium sextuple {ϕ∗, ϕ∗∗, P1, P2, R1, R2} defines unique
equilib-rium values of all other endogenous variables of the model {M1, M2, Me,
LP1, LP2, Le, C1, C2}
Proof. See Appendix B
We have fully characterized the equilibrium sextuple {ϕ∗, ϕ∗∗, P1, P2,
R1, R2} and solved for the other endogenous variables of the model. The
firms, heterogeneous products, and industry dynamics in the form of ongo-ing entry and exit. In equilibrium, firm productivity is correlated with product-level fixed costs, with the most productive firms endogenously choosing to make products with the highest fixed costs.
6. Market Conditions and Product Choice
In this section, we examine how changes in market conditions influence firm product choice and result in product switching. Key product char-acteristics and features of market structure are: the sunk costs of entry (fe), the fixed costs of production (f1,f2), the variable cost of production
for product 2 relative to that for product 1 (b), and the demand-shifter (a) capturing the relative weight of the two products in consumers’ utility.
Until now we have not restricted the distribution of firm productivity, g(ϕ). In this section we focus on the results assuming that the productivity distribution g(ϕ) is Pareto with parameters a and k: g(ϕ) = akaϕ−(a+1), where k > 0, a > 0, ϕ ≥ k, and G(ϕ) = 1 −³kϕ
´a
.16 This assumption simplifies the analysis, as demonstrated in Appendix B, which also contains formal derivations of the comparative statics in this section.
In the interests of brevity, we provide a complete analysis of one aspect of market structure - the sunk costs of entry (fe). These may be thought
of as capturing barriers to entry in the industry, and may be of particular interest in so far as they can be directly influenced by policy. The impact of changes in other market conditions are analogous. We end the section with a more general discussion of how other parameters of the model influence product switching.
As shown in Table 5, an increase in the sunk costs of entry in the industry (fe) lowers both cutoff productivity levels thus decreasing average
productivity in each product and for the industry as a whole. The ratio of the productivity cutoffs, the relative price of the products, the mass of firms producing each product, and average profitability are unchanged. The expected value of entry rises and welfare unambiguously falls.
1 6
The distribution of within industry (SIC4) cross-firm labor productivity in the US is well-approximated by a Pareto distribution (formally a Pareto-1 distribution). Compar-ative statics without assuming a particular distribution for firm productivity are available upon request from the authors.
Table 5: Comparative Statics for a Change in the Sunk Cost of Entry ∂ϕ∗ ∂fe
< 0
∂ϕ∗∗ ∂fe< 0
∂(ϕ∗∗/ϕ∗) ∂fe= 0
∂P ∂fe= 0
∂ϕ(ϕ∗,ϕ∗∗) ∂fe< 0
∂ϕ(ϕ∗,ϕ∗∗) ∂fe< 0
∂π1 ∂fe= 0
∂M1 ∂fe= 0
∂π2 ∂fe= 0
∂M2 ∂fe= 0
∂ve ∂fe> 0
∂W ∂fe< 0
As the sunk costs of entry rise above the expected value of entry, a smaller mass of firms, Me, will enter the industry. For given values of ϕ∗
and ϕ∗∗, a smaller mass of entrants implies a smaller mass of firms with productivity realizations high enough to produce in each market. This fall in the mass of firms producing in each market increases ex post profitability. The increase in ex post profitability means that firms with lower re-alizations of productivity than before are able to cover the fixed costs of producing product 1. Hence, in equilibrium the zero profit cutoff produc-tivity ϕ∗ falls. As ϕ∗ falls for a given value of ϕ∗∗, this increases the mass
of firms in product 1 relative to the mass of firms in product 2, thereby reducing product 1’s relative profitability. Hence, some previously high productivity manufacturers of product 1 now find it more profitable to produce the high fixed cost product 2 and ϕ∗∗ also falls.
The equilibrium ratio of the two productivity cutoffs, ϕ∗∗/ϕ∗, is inde-pendent of the sunk costs of entry, and hence ϕ∗∗ falls by the same
propor-tion as ϕ∗. With a Pareto productivity distribution, this leaves the relative price of the two products, P, unchanged.
The fall in both ϕ∗ and ϕ∗∗ means that some low productivity firms
productivity manufacturers of product 1 now produce product 2. For both reasons, weighted average productivity in product 1,ϕ, will fall. Similarly,e the fall in ϕ∗∗ means that product 2 now includes some lower productivity firms who previously manufactured product 1. Hence, weighted average productivity in product 2, eeϕ, will also fall.
The fall in ϕ∗ and ϕ∗∗ increases the mass of firms with productivity realizations high enough to produce in each market for a given mass of firms, Me, that enter. With a Pareto distribution, this effect exactly
offsets the smaller mass of firms entering the industry, so that the mass of firms producing in each product market (M1, M2) and average ex post
profitability (π1,π2) are unchanged.
The expected value of entry, ve, rises to equal the new higher sunk
costs of entry, fe, because the fall in ϕ∗ and ϕ∗∗ increases the expected
probability of a firm having a productivity realization high enough to be able to profitably manufacture either product 1 or product 2. Welfare per worker, W , falls because, although the mass of firms and hence product varieties is unchanged, the fall in average productivity within each product market leads to a rise in average prices.
The effects of changes in other market conditions on product choice are formalized in Appendix B. Increases in the fixed production cost for prod-uct 2 (f2) reduce relative profitability in this product market, increasing
the relative mass of firms that make product 1, and leading to a rise in ϕ∗∗/ϕ∗. Increases in the fixed production cost for product 1 (f1) have
exactly the opposite effect, increasing the relative mass of firms that make product 2 and reducing ϕ∗∗/ϕ∗.
Increases in the variable production cost for product 2 (b) reduce relative profitability for this product, again increasing the relative mass of firms that make product 1, and leading to a rise in ϕ∗∗/ϕ∗. Increases in the weight of product 1 in consumers’ utility (a) raise the relative demand for this product, increasing the relative mass of firms that make product 1, and again leading to a rise in ϕ∗∗/ϕ∗.
In each comparative static, market conditions affect both firms’ incen-tives to enter and exit the market (ϕ∗) and to choose between the two
products (ϕ∗∗). Changes in product characteristics and market structure result in systematic product switching, with the interaction between firm heterogeneity and product characteristics shaping the pattern of product
switching. When profitability rises in the market for the high fixed cost product 2, it is the more productive firms within product product 1 who switch towards this product. Similarly, when profitability rises in the mar-ket for the low fixed cost product 1, it is the less productive firms within product 2 who switch.
7. Conclusions
This paper develops a model with endogenous product selection in light of new evidence on the importance of product switching by manufacturing firms. Two thirds of continuing firms change their product mix in any five year period. Multiple-product firms that both drop and add products account for over three quarters of total output. On average, product switches affect more than 40% of firm output and almost half of existing products. Product additions also involve important changes in the industry composition of the firm, extending the range of industries in more than half of all cases.
Motivated by these stylized facts, the paper develops a theoretical model that integrates endogenous product choice into a dynamic analysis of in-dustry evolution with entry and exit and heterogeneous firms. In equilib-rium, firm productivity is correlated with product-level fixed costs, with the most productive firms endogenously choosing to make the products with the highest fixed costs. Changes in market structure result in systematic patterns of firm entry/exit and product switching by continuing firms. For example, increases in an industry’s sunk costs of entry make it profitable for less productive firms to survive in the market; result in product switch-ing as the more productive producers of the low fixed cost product 1 switch to the high fixed cost product 2; and induce a fall in average productivity within product markets.
The theoretical model introduces endogenous product choice and the possibility of product switching by surviving firms. However, the scope of product switching in the data suggests that future work should allow for firm characteristics, such as productivity, that evolve over time. The addi-tion of shocks to the firm productivity process would provide an addiaddi-tional source of product switching. A multi-industry version of the model would allow for product-switching that changes a firm’s industry mix.
There are a number of theoretical predictions of the current model that are amenable to empirical testing including the response of product switch-ing to changes in market structure such as the sunk cost of entry. A particularly interesting area for further theoretical research is the intro-duction of international trade. Existing research suggests a number of ways in which firms in developed countries respond to increased global-ization: the death of less productive firms; entry into exporting by high productivity firms; and changes in industry composition as more produc-tive firms expand. More recent empirical research has provided evidence of firms switching industry in response to increased competition from low wage countries. This paper suggests that another important margin along which firms may adjust to increased globalization and other changes in the competitive structure of markets is through product choice and/or changes in the nature of the production process.
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A Appendix A: Data
The data comes from the Censuses of Manufactures (CM) of the Longi-tudinal Research Database (LRD) of the U.S. Bureau of the Census starting in 1972 and conducted every fifth year through 1997. The sampling unit for the Census is a manufacturing establishment, or plant. Establishments that are not mailed a Census form, so-called ‘administrative records (AR)’, do not report product-level data. Any firm-year observation that includes an AR establishment is excluded from the sample. Product additions and deletions are recorded only for firms that are present in the sample in neigh-boring censuses, i.e. we exclude firm births and deaths as well as changes from AR status to regular Census status to focus on product switching at suriving firms. Plant acquisitions and sales by firms will potentially result in changes in product mix, although they account for fewer than 10% of such switches in the sample.
In Census years, plant output is recorded either at the five-digit or seven-digit SIC level of detail. Roughly 7,000 of the 15,000 seven-digit categories are recorded directly in the LRD, the rest are recorded at a five-digit level of aggregation. We aggregate seven-digit categories up to their five-digit ‘product-class’ to obtain a comparable level of product detail for all manufacturing firms. A small fraction of firm-product-year observations report negative values for output. We recode these values to zero but leave the products in the firm mix (none of the results are sensitive to this choice).
For the most part, five-digit 1987 SIC categories correspond to prod-ucts that are likely to be imperfect substitutes and made by different pro-duction processes. For example, “Nonferrous Wiredrawing and Insulat-ing” (SIC 3357) includes “Copper Wire”, “Power Wire and Cable”, and “Fiber Optic Cables”. Similarly, “Motor Vehicles” (SIC 3711) contains seven five-digit categories including “Passenger Cars”, “Buses”, “Com-bat Vehicles”, and four categories of “Trucks” distinguished by weight. Adding or dropping one of these products is likely to entail a substan-tial change in the production process. The SIC system does not always categorize products into five-digit codes that represent such distinct pro-duction processes. "Canned Fruits and Vegetables" (SIC 2033) contains two five-digit products, “Canned Fruits” and “Canned Vegetables, Except Mushrooms”. Switches between these products are unlikely to involve
changes in production technique.
B Appendix B: Theoretical Derivations
This theoretical appendix contains detailed derivations of key results used in the text and proofs of Propositions.
B1. Product Choice, Ex Ante, and Ex Post Productivity Distributions The ex ante distribution of firm productivity prior to entry is g(ϕ). The analysis of product choice in the main text implies that firms’ ex ante probability of successful entry is [1 − G (ϕ∗)], where G (·) is the cumulative
distribution of g (·). The ex ante probability of producing the low quality good is [G (ϕ∗∗) − G (ϕ∗)], and the ex ante probability of producing the high quality good is [1− G (ϕ∗∗)].
Similarly, the analysis of product choice implies that the ex post distri-bution of firm productivity for product 1, µ1(ϕ), is the conditional distrib-ution of g (ϕ) on [ϕ∗, ϕ∗∗),
µ1(ϕ) =
( g(ϕ)
G(ϕ∗∗)−G(ϕ∗) if ϕ∗ ≤ ϕ < ϕ∗∗
0 otherwise (24)
while the ex post distribution of firm productivity for product 2, µ2(ϕ), is the conditional distribution of g (ϕ) on [ϕ∗∗, ∞),
µ2(ϕ) =
( g(ϕ)
1−G(ϕ∗∗) if ϕ∗∗≤ ϕ < ∞
0 otherwise . (25)
B2. Aggregate Price Indices
Equilibrium is characterized by constant masses of firms, M1 and M2,
producing the two products (and hence M1 and M2 varieties of the
prod-ucts). Aggregate price indices may be written as:
P1 = ·Z ∞ 0 p1(ϕ)1−σM1µ1(ϕ) dϕ ¸1/1−σ (26) P2 = ·Z ∞ 0 p2(ϕ)1−σM2µ2(ϕ) dϕ ¸1/1−σ . (27)
Using the equilibrium pricing rule (6), the price index for each product may be written as proportional to the price charged by a firm with the (weighted) average productivity of all firms producing that product (equation (16) in the main text). Weighted average productivity is defined as:
e ϕ (ϕ∗, ϕ∗∗) = " 1 G (ϕ∗∗) − G (ϕ∗) Z ϕ∗∗ ϕ∗ ϕσ−1g (ϕ) dϕ #1/(σ−1) (28) eeϕ (ϕ∗, ϕ∗∗) = · 1 1 − G (ϕ∗∗) Z ∞ ϕ∗∗ ϕσ−1g (ϕ) dϕ ¸1/(σ−1) (29)
where ϕ(·) and ee ϕ(·) are weighted harmonic means of the ϕ’s; the weightse reflect relative output shares of firms with different levels of productivi-ties. Note that these weighted averages depend solely on the two cutoff productivities (ϕ∗, ϕ∗∗) and the ex ante distribution g(ϕ).
B3. Average Profitability
This section of the appendix establishes two results. First, expected or average profits in each market (π1, π2) are equal to the profits earned by a
firm with weighted average productivity (eϕ, feϕ). Second, the profits earned by a firm with weighted average productivity may be written as functions of the productivity cutoffs and relative prices (ϕ∗, ϕ∗∗, P).
Taking averages in the profit function (7) and using the definitions of weighted average productivity (ϕ, ee ϕ), average profit in each market is equale to the profit of a firm with weighted average productivity:
π1 = π1(ϕ),e π2= π2(eϕ).e
The relationships between the revenues received by firms with different levels of productivity (equations (8) and (9)) imply that the revenue of a firm with weighted average productivity in each market can be expressed relative to the revenue of a firm with the zero profit cutoff level of produc-tivity ϕ∗. From the zero profit condition (ZP), the revenue of a firm with the zero profit cutoff level of productivity is proportional to the fixed cost: r1(ϕ∗) = σf1
Average profit in the two markets is thus equal to: π1(ϕ∗, ϕ∗∗) = "µ e ϕ (·) ϕ∗ ¶σ−1 − 1 # f1 (30) π2(ϕ∗, ϕ∗∗, P) = µ1 − a a ¶ψà 1 b eeϕ (·) ϕ∗ !σ−1 Pσ−ψ−ff2 1 f1 (31)
where (ϕ, ee ϕ) depend solely on the productivity cutoffs (ϕe ∗, ϕ∗∗), and we have used the expressions for CES expenditure shares, αi(P), in equation
(5) to simplify terms.
B4. Product Supply and Relative Prices The conditions ZP and PI imply,
r1(ϕ∗) = σf1 (32)
r2(ϕ∗∗) = r1(ϕ∗∗) + σ(f2− f1). (33)
The relationship between the revenues received by firms of different pro-ductivity in product 1 (equation 8) implies,
r1(ϕ∗∗) = µ ϕ∗∗ ϕ∗ ¶σ−1 r1(ϕ∗) (34) r2(ϕ∗∗) = µ 1 − a a ¶ψµ 1 b ¶σ−1 Pσ−ψr1(ϕ∗∗) . (35)
Combining these four equations, we obtain equation (20) in the main text.
B5. Product Demand and Relative Prices
The relative price of product 2, P, is the ratio of the price indices in equation (16) and depends on the relative mass of firms producing the two products, M2/M1. From the steady-state stability conditions SC, the
equilibrium relative mass of firms producing the two products is: M2
M1
= [1 − G(ϕ
∗∗)]
Combining this with the definitions of weighted average productivity (eϕ, eeϕ) in equations (28) and (29), P may be written as the following function of the productivity cutoffs ϕ∗∗ and ϕ∗:
P = b "Rϕ∗∗ ϕ∗ ϕσ−1g (ϕ) dϕ R∞ ϕ∗∗ϕσ−1g (ϕ) dϕ # 1 σ−1 . (36)
which is equivalent to equation (22) in the main text.
B6. Combining Product Supply and Product Demand
It will prove convenient to rearrange the product supply relationship (20) so that P appears on the left-hand side:
P = bσσ−ψ−1 µ a 1 − a ¶ ψ σ−ψ "µϕ∗∗ ϕ∗ ¶1−σµf 2 f1 − 1 ¶ + 1 # 1 σ−ψ . (37)
Since σ > 1, the right-hand side is monotonically decreasing in ϕ∗∗/ϕ∗
and is graphed in (P, ϕ∗∗/ϕ∗) space in Figure 2. P takes the value (f2/f1)1/(σ−ψ)(a/(1 − a))ψ/(σ−ψ)b(σ−1)/(σ−ψ) > 0 at ϕ∗∗/ϕ∗ = 1 and
con-verges to a lower value of (a/(1 − a))ψ/(σ−ψ)b(σ−1)/(σ−ψ) > 0 as ϕ∗∗/ϕ∗ tends to infinity.
The product demand relationship (22) has already been rearranged so that P appears on the left-hand side in equation (36) above. The right-hand side of equation (36) is monotonically increasing in ϕ∗∗/ϕ∗ and is also graphed in (P, ϕ∗∗/ϕ∗) space below. As ϕ∗∗/ϕ∗ approaches 1, P converges to 0. As ϕ∗∗/ϕ∗ tends to infinity, P converges to ∞.
It follows that there exists a unique equilibrium value of ϕ∗∗/ϕ∗ > 1 where both the product supply and product demand conditions are satis-fied, as shown graphically in Figure 2. From the product supply condition, this implies a value for the equilibrium relative price, P, such that the inequality in equation (21) is satisfied and both products are produced.
B7. Free Entry
Substitute the expressions for average profit in equations (30) and (31) into the free entry condition FE. Substitute for ϕ∗∗ in FE using (20).
0 1 1 2 P P * * * ϕ ϕ RD RS 1 2 ˆ P P * * * ˆ ϕ ϕ ψ σσ ψ σ ψ ψ σ − − − − − 1 1 1 2 1 a b a f f ψ σ σ ψ σψ − − − − 1 1 a b a
Figure 2: Equilibrium P and ϕ∗∗/ϕ∗
Use the definitions of weighted average productivity (ϕ, ee ϕ) in equationse (28) and (29) to simplify terms. As a result, FE may be written in terms of ϕ∗, Λ ≡ ϕ∗∗/ϕ∗ and P as in equation (23) in the main text.
B8. Proof of Proposition 1
Proof. (a) Equations (20) and (22) define unique equilibrium values of (ϕ∗∗/ϕ∗, P) as demonstrated in section B6. above. Given values of
